{
  "id": "1105.3410",
  "version": "v3",
  "published": "2011-05-17T15:10:20.000Z",
  "updated": "2012-11-08T10:48:55.000Z",
  "title": "Lagrangian fibrations on hyperkähler manifolds - On a question of Beauville",
  "authors": [
    "Daniel Greb",
    "Christian Lehn",
    "Sönke Rollenske"
  ],
  "comment": "28 pages; v2: minor corrections, added Thm 6.7; v3: implemented referees' comments, added slightly more detailed argument in Sect. 6.3.3, accepted for publication by Annales scientifiques de l'\\'Ecole normale sup\\'erieure",
  "journal": "Ann. Sci. \\'Ec. Norm. Sup\\'er. (4) 46 (2013), no. 3, 375-403",
  "categories": [
    "math.AG",
    "math.CV",
    "math.DG"
  ],
  "abstract": "Let X be a compact hyperk\\\"ahler manifold containing a complex torus L as a Lagrangian subvariety. Beauville posed the question whether X admits a Lagrangian fibration with fibre L. We show that this is indeed the case if X is not projective. If X is projective we find an almost holomorphic Lagrangian fibration with fibre L under additional assumptions on the pair (X, L), which can be formulated in topological or deformation-theoretic terms. Moreover, we show that for any such almost holomorphic Lagrangian fibration there exists a smooth good minimal model, i.e., a hyperk\\\"ahler manifold birational to X on which the fibration is holomorphic.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2012-11-08T10:48:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C26",
      "14D06",
      "14E30",
      "32G10",
      "32G05"
    ],
    "keywords": [
      "hyperkähler manifolds",
      "holomorphic lagrangian fibration",
      "complex torus",
      "lagrangian subvariety",
      "manifold birational"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1105.3410G"
    }
  }
}